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Mathematics > Combinatorics

Title:Coding Theory and Uniform Distributions

Abstract: In the present paper we introduce and study finite point subsets of a special
kind, called optimum distributions, in the n-dimensional unit cube. Such
distributions are closely related with known (delta,s,n)-nets of low
discrepancy. It turns out that optimum distributions have a rich combinatorial
structure. Namely, we show that optimum distributions can be characterized
completely as maximum distance separable codes with respect to a non-Hamming
metric. Weight spectra of such codes can be evaluated precisely. We also
consider linear codes and distributions and study their general properties
including the duality with respect to a suitable inner product. The
corresponding generalized MacWilliams identities for weight enumerators are
briefly discussed. Broad classes of linear maximum distance separable codes and
linear optimum distributions are explicitely constructed in the paper by the
Hermite interpolations over finite fields.